The first thing I'd mention is that 3/4 of the people you ask will tell you that wherever they live is great, that they couldn't see themselves living anywhere else etc etc, and they will blatantly lie on their commute time, which probably takes twice as long as what they'll tell you. I lived abroad and didn’t notice this as much abroad, whereas here I have NEVER, not once, heard anyone saying: “I moved to X because I couldn’t afford Y; the commute sucks but at least I have managed to afford a nice house”. Anyway…
What's your budget? Not just to rent but also to buy, unless you are happy renting somewhere now then moving to a different area to buy?
Where do you work, what are your transport needs?
Do you have a car?
Do you want an area where you can do without one?
How far west do you want to consider? Up to Putney or also Sheen Richmond etc?
My two cents is this: In all these areas I would avoid newbuilds like the plague. Newbuild flats, even if with garden, are too expensive vs older houses, come with huge management charges over which you have no control (and which could therefore increase at a crazy pace), and are generally not what British families look for. If you look up sold prices on right move you’ll see many newbuild flats have been sold either at a loss or at a much smaller gain than older houses (eg a flat appreciated by 3% when houses appreciated by 15%).
A good thing about the train services in most of these areas is that they are very frequent, and you can often go to the station as if it were the tube, ie without even checking the timetable. This can make a big difference, because if you live in an area where trains are every 10 minutes, if one train gets cancelled and the next delayed, it starts to add up. Also, there are many train lines that converge at Wimbledon or Balham, so there is an element of redundancy: if one line has a problem you still have other lines which are running. If you live somewhere with only one line, good luck for your commute.
As for areas: in general, all the areas below have good access to parks. You need to think if you want an area from which you don’t need a car (Balham and Wimbledon are better than Southfields and Earslfield for this), and which transport link is more important (district line, train to Waterloo, train to Victoria).
Southfields and Earlsfield: the locals will feel offended, but I dare think the high streets suck - there is little to nothing, not even a big supermarket (you need a car to go to one) BUT there is a lot nearby, from Wimbledon, which you can reach by public transport, to the Southside shopping center, for which you most likely need a car. Whether the Earlsfield train to Clapham Junction and Waterloo or the Southfields district line is more useful depends on where you need to go. That branch of the district line is neither fast nor frequent, so if you have to get to, say, the City, it might actually be quicker from Earlsfield. If you have to get to Earls court - Chelsea then the district line is better.
Wimbledon: the best of both world with both the district line and the train. Very lively in terms of shops restaurants etc (at least pre-covid). I like the neighbourhood but find houses a bit too expensive for my budget. Some people don't like the fact that it's farther away from the centre.
Balham: the high street has a bit more than Southfields and Earslfield but there is less nearby. You can get to big supermarkets without a car but shopping centres are farther away. Having both the tube and the train is good but the tube used to get extremely crowded at rush hour. Balham is on the line to Victoria, not to Waterloo like Wimbledon and Earslfield - which is more useful depends on where you need to go.
Putney: being close to the river is great, Wandsworth park is very nice, the high street is nice and lively, but the proximity to a river crossing means the area is always busy with traffic, especially since they closed Hammersmith bridge (and it doesn't seem like it will be reopened any time soon). It has train and tube.
Battersea park: I personally don't like it. The park is great, but the new developments cost crazy money, and older houses are too expensive for what you get, because there are many dodgy areas around and because transport options are very limited for now. Most trains pass by Battersea park station without stopping, and busses take forever.
Wandsworth common: beautiful park, but non-existent high-street, and, like Battersea park, most trains pass by without stopping.
Wandsworth town: limited high street, similar to Earlsfield and Southfields, good transport options by train, access to parks slightly worse than from other areas (unless you count Bramsford gardens).
Clapham common / Clapham North: maybe more popular with childless people in their 20s and 30s than with families. Transport isn't great because the tube used to get incredibly busy pre-covid.
Clapham Junction / Northcote rd: Clapham Junction station has excellent transport links, with very frequent trains to Victoria and to Waterloo. The Northcote rd high street is, I find, very nice, and is between the commons, but if you want to be close to Northcote rd and the local schools, you are probably at least 15 minutes away on foot from the station. If you are closer to the station you are farther from the schools...